Empirical Likelihood for Outlier Detection and Estimation in Autoregressive Time Series

Identification and estimation of outliers in time series is proposed by using empirical likelihood methods. Theory and applications are developed for stationary autoregressive models with outliers distinguished in the usual additive and innovation types. Some other useful outlier types are considered as well. A simulation experiment is used for studying the behaviour of the empirical likelihood‐based method in finite samples and indicates that the proposed methods are preferable when dealing with the non‐Gaussian data. Our simulations suggest that the usual sequential procedure for multiple outlier detection is suitable also for the methods based on empirical likelihood.     

一种改进的动态数据校正离群值检测法

Data reconciliation technology can decrease the level of corruption of process data due to measurement noise, but the presence of outliers caused by process peaks or unmeasured disturbances will smear the reconciled results. Based on the analysis of limitation of conventional outlier detection algorithms, a modified outlier detection method in dynamic data reconciliation （DDR） is proposed in this paper. In the modified method, the outliers of each variable are distinguished individually and the weight is modified accordingly. Therefore, the modified method can use more information of normal data, and can efficiently decrease the effect of outliers. Simulation of a continuous stirred tank reactor （CSTR） process verifies the effectiveness of the proposed algorithm.     

Outlier detection in ARMA models

Abstract. We consider an autoregressive moving‐average (ARMA) time series where the observations are perturbed by two kinds of outliers: an additive outlier (AO) or an innovation outlier (IO). Abraham and Yatawara [Journal of Time Series Analysis (1988) Vol. 9, pp. 109–19] investigate a sequential test which successively detects and identifies the outlier type. In this article, we propose an extension of this test, called ‘modified sequential test’, which performs the two procedures simultaneously and coherently. The asymptotic distribution of the test statistic is calculated under the null hypothesis that no outlier is present. Comparison of the two test procedures using simulation experiments shows that the proposed test gives a better power especially in the case of an IO.     

钢铁企业高炉煤气发生量异常数据检测

从高炉煤气生产的实际工况出发,对异常数据产生的原因和特点进行分析。针对现有异常检测方法运算效率低下的问题,提出一种改进的局部异常因子检测算法。该算法首先利用五数总括法剔除掉大量的正常数据,然后再用一种相对k距离的比值表示剩余离群点的异常程度,进而判断异常数据。仿真实验表明:改进方法检测所需的时间比传统的局部异常因子方法检测所需的时间更少,且检测效果更加准确、直观。     

一种新型融合离群点的稳态检测方法

针对带有离群点的数据稳态检测,采用分布图法对离群点进行剔除;为了保持数据的完整性,提出用灰色预测值替代离群点值;最后用3δ法则进行稳态检验。如此,数据的稳态与非稳态便会区分开来。与现有稳态检测方法相比,分布图法快速有效地克服了离群点对稳态检测结果不准确的影响,降低了过程中个别异常数据带来的误诊率;灰色预测方法使离群点的替代值更贴近真实值,从而得到的过程数据比现有方法得到的数据更可靠。仿真结果证实了该方法的有效性和优越性。     

Industrial melt index prediction with the ensemble anti‐outlier just‐in‐time Gaussian process regression modeling method

Several data‐driven soft sensors have been applied for online quality prediction in polymerization processes. However, industrial data samples often follow a non‐Gaussian distribution and contain some outliers. Additionally, a single model is insufficient to capture all of the characteristics in multiple grades. In this study, the support vector clustering (SVC)‐based outlier detection method was first used to better handle the nonlinearity and non‐Gaussianity in data samples. Then, SVC was integrated into the just‐in‐time Gaussian process regression (JGPR) modeling method to enhance the prediction reliability. A similar data set with fewer outliers was constructed to build a more reliable local SVC–JGPR prediction model. Moreover, an ensemble strategy was proposed to combine several local SVC–JGPR models with the prediction uncertainty. Finally, the historical data set was updated repetitively in a reasonable way. The prediction results in the industrial polymerization process show the superiority of the proposed method in terms of prediction accuracy and reliability. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015 , 132, 41958.     

SEARCHING FOR ADDITIVE OUTLIERS IN NONSTATIONARY TIME SERIES*

Abstract. Recently, Vogelsang (1999) proposed a method to detect outliers which explicitly imposes the null hypothesis of a unit root. It works in an iterative fashion to select multiple outlier in a given series. We show, via simulations, that, under the null hypothesis of no outliers, it has the right size in finite samples to detect a single outlier but, when applied in an iterative fashion to select multiple outliers, it exhibits severe size distortions towards finding an excessive number of outliers. We show that his iterative method is incorrect and derive the appropriate limiting distribution of the test at each step of the search. Whether corrected or not, we also show that the outliers need to be very large for the method to have any decent power. We propose an alternative method based on first‐differenced data that has considerably more power. We also show that our method to identify outliers leads to unit root tests with more accurate finite sample size and robustness to departures from a unit root. The issues are illustrated using two US/Finland real‐exchange rate series.     

State space models for time series with patches of unusual observations

Abstract. An alternative to leave‐k‐out diagnostics for detecting patches of outlying points in time series is developed. We propose that unusual behaviour should be modelled by the addition of shocks. By including shocks in the transition equation of a state space model, we admit the possibility of a persistent change associated with a patch of outliers. Persistent change may take the form of a level shift or a change in seasonal pattern. We provide an efficient mechanism for computing diagnostic statistics associated with the addition of k shocks using a simple adaptation of the Kalman filter. Statistics for detecting unspecified patterns of shocks and an interpretation of the output of the associated smoothing algorithm are derived. Illustrations using real series are given.     

Bayesian Outlier Detection in Non‐Gaussian Autoregressive Time Series

This work investigates outlier detection and modelling in non‐Gaussian autoregressive time series models with margins in the class of a convolution closed parametric family. This framework allows for a wide variety of models for count and positive data types. The article investigates additive outliers which do not enter the dynamics of the process but whose presence may adversely influence statistical inference based on the data. The Bayesian approach proposed here allows one to estimate, at each time point, the probability of an outlier occurrence and its corresponding size thus identifying the observations that require further investigation. The methodology is illustrated using simulated and observed data sets.     

Ensemble Correntropy‐Based Mooney Viscosity Prediction Model for an Industrial Rubber Mixing Process

The development of accurate soft sensors for online prediction of Mooney viscosities in industrial rubber mixing processes is a difficult task because the modeling dataset often contains various outliers. A correntropy kernel learning (CKL) method for robust soft sensor modeling of nonlinear industrial processes with outlier samples is proposed. Simultaneously, the candidate outliers can be identified once the CKL‐based soft sensor model is built. An index for describing the uncertainty of the CKL model is designed. Furthermore, to obtain more robust and accurate predictions, an ensemble CKL (ECKL) method is formulated by introducing the simple bagging strategy. Consequently, by detecting the outliers in a sequential manner, the database becomes more reliable for long‐term use. The application results for the industrial rubber mixing process demonstrate the superiority of ECKL in terms of better prediction performance.     

A SCORE TEST FOR DETECTION OF TIME SERIES OUTLIERS

Abstract. Two characterizations, the aberrant observation and innovation models, for outliers in time series are considered. A procedure based on the well-known score-test is discussed for detection of outliers and distinguishing between the outlier types. Significance levels of the tests are also obtained and the method is illustrated with simulated examples.     

Nonlinear stochastic modeling to improve state estimation in process monitoring and control

State estimation from plant measurements plays an important role in advanced monitoring and control technologies, especially for chemical processes with nonlinear dynamics and significant levels of process and sensor noise. Several types of state estimators have been shown to provide high‐quality estimates that are robust to significant process disturbances and model errors. These estimators require a dynamic model of the process, including the statistics of the stochastic disturbances affecting the states and measurements. The goal of this article is to introduce a design method for nonlinear state estimation including the following steps: (i) nonlinear process model selection, (ii) stochastic disturbance model selection, (iii) covariance identification from operating data, and (iv) estimator selection and implementation. Results on the implementation of this design method in nonlinear examples (CSTR and large dimensional polymerization process) show that the linear time‐varying autocovariance least‐squares technique accurately estimates the noise covariances for the examples analyzed, providing a good set of such covariances for the state estimators implemented. On the estimation implementation, a case study of a chemical reactor demonstrates the better capabilities of MHE when compared with the extended Kalman filter. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Robust modeling of mixture probabilistic principal component analysis and process monitoring application

In this article, a robust modeling strategy for mixture probabilistic principal component analysis (PPCA) is proposed. Different from the traditional Gaussian distribution driven model such as PPCA, the multivariate student t‐distribution is adopted for probabilistic modeling to reduce the negative effect of outliers, which is very common in the process industry. Furthermore, for handling the missing data problem, a partially updating algorithm is developed for parameter learning in the robust mixture PPCA model. Therefore, the new robust model can simultaneously deal with outliers and missing data. For process monitoring, a Bayesian soft decision fusion strategy is developed which is combined with the robust local monitoring models under different operating conditions. Two case studies demonstrate that the new robust model shows enhanced modeling and monitoring performance in both outlier and missing data cases, compared to the mixture probabilistic principal analysis model. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2143–2157, 2014     

THE DETECTION OF A SINGLE ADDITIVE OUTLIER OF UNKNOWN POSITION

Abstract. Kudo (On the testing of outlying observations. Sankhya 17 (1956), 67–73) has derived an optimal invariant detector of a single additive outlier of unknown position in the context of an underlying Gaussian process consisting of independent and identically distributed random variables. We show how this author's arguments can be extended to derive an invariant detector of an additive outlier of unknown position for an underlying zero-mean Gaussian stochastic process. This invariant detector depends on the parameters of this process; its properties are analysed further for the particular case of an underlying zero-mean Gaussian AR( p ) process. It provides an upper bound on the performance of any invariant detector based solely on the data and it may be 'bootstrapped' to provide an invariant detector based solely on the data. A plausibility argument is presented in favour of the proposition that the bootstrapped detector is nearly optimal for sufficiently large data length n. The truth of this proposition has been confirmed by simulation results for zero-mean Gaussian AR(1) and AR(2) processes (for certain sets of possible outlier positions). The bootstrapped detector is shown to be closely related to the detector based on the approximate likelihood ratio criteria of Fox (Outliers in time series. J. Roy. Statist. Soc. Ser. B 34 (1972), 350–63) and the leave-one-out diagnostic of Bruce and Martin (Leave- k -out diagnostics in time series. J. Roy. Statist. Soc. Ser B 51 (1989), 363–424). It is also shown how the case of an underlying Gaussian process with arbitrary mean can be reduced to the case of an underlying zero-mean Gaussian process.     

Time‐series models with an EGB2 conditional distribution

A time‐series model in which the signal is buried in noise that is non‐Gaussian may throw up observations that, when judged by the Gaussian yardstick, are outliers. We describe an observation‐driven model, based on an exponential generalized beta distribution of the second kind (EGB2), in which the signal is a linear function of past values of the score of the conditional distribution. This specification produces a model that is not only easy to implement but which also facilitates the development of a comprehensive and relatively straightforward theory for the asymptotic distribution of the maximum‐likelihood (ML) estimator. Score‐driven models of this kind can also be based on conditional t distributions, but whereas these models carry out what, in the robustness literature, is called a soft form of trimming, the EGB2 distribution leads to a soft form of Winsorizing. An exponential general autoregressive conditional heteroscedastic (EGARCH) model based on the EGB2 distribution is also developed. This model complements the score‐driven EGARCH model with a conditional t distribution. Finally, dynamic location and scale models are combined and applied to data on the UK rate of inflation.     

Robustness of Zero Crossing Estimator

Zero crossing (ZC) statistic is the number of zero crossings observed in a time series. The expected value of the ZC specifies the first‐order autocorrelation of the processes. Hence, we can estimate the autocorrelation by using the ZC estimator. The asymptotic consistency and normality of the ZC estimator for scalar Gaussian processes are already discussed in 1980. In this article, first, we derive the joint asymptotic distribution of the ZC estimator for ellipsoidal processes. Next, we show the variance of the ZC estimator does not attain the Cramer–Rao lower bound (CRLB). However, it is shown that the ZC estimator has robustness when the process is contaminated by an outlier. In contrast with this, we observe that the quasi‐maximum likelihood estimator (QMLE) attains the CRLB. However, we can see that QMLE is sensitive for the outlier.     

Range Unit‐Root (RUR) Tests: Robust against Nonlinearities,Error Distributions,Structural Breaks and Outliers

Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit‐root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long‐wave patterns observed not only in unit‐root time series but also in series following more complex data‐generating mechanisms. To this end, our testing device analyses the unit‐root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit‐root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit‐root tests on near‐unit‐root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward–backward range unit‐root (FB‐RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey–Fuller unit‐root test on exchange rate series.     

Monitoring and control of styrene solution polymerization using NIR spectroscopy

Tailored polymer resins are frequently required for a given application. The lack of instruments for in‐line monitoring of polymer quality has long been recognized as an important problem in polymerization reactor control. Using the styrene solution polymerization system as an example, we present the use of near‐infrared (NIR) spectroscopy as an alternative tool for in‐line and in situ monitoring and control of monomer conversion and average molecular weight of polymer resins. By using a Kalman filter state estimator and an accurate first‐principle model, the control loop could be successfully closed to track desired monomer values and average molecular weights. Two process control strategies, one based on the optimal control theory and the other on model predictive control, were implemented both theoretically and experimentally. The experimental results showed that it is feasible to use NIR spectroscopy for the simultaneous control of monomer conversion and polymer average molecular weight. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 90: 1273–1289, 2003     

Interventions in INGARCH processes

We study the problem of intervention effects generating various types of outliers in a linear count time‐series model. This model belongs to the class of observation‐driven models and extends the class of Gaussian linear time‐series models within the exponential family framework. Studies about effects of covariates and interventions for count time‐series models have largely fallen behind, because the underlying process, whose behaviour determines the dynamics of the observed process, is not observed. We suggest a computationally feasible approach to these problems, focusing especially on the detection and estimation of sudden shifts and outliers. We consider three different scenarios, namely the detection of an intervention effect of a known type at a known time, the detection of an intervention effect when the type and the time are both unknown and the detection of multiple intervention effects. We develop score tests for the first scenario and a parametric bootstrap procedure based on the maximum of the different score test statistics for the second scenario. The third scenario is treated by a stepwise procedure, where we detect and correct intervention effects iteratively. The usefulness of the proposed methods is illustrated using simulated and real data examples.     

Robust discrimination between long‐range dependence and a change in mean

In this article we introduce a robust to outliers Wilcoxon change‐point testing procedure, for distinguishing between short‐range dependent time series with a change in mean at unknown time and stationary long‐range dependent time series. We establish the asymptotic distribution of the test statistic under the null hypothesis for L₁ near epoch dependent processes and show its consistency under the alternative. The Wilcoxon‐type testing procedure similarly as the CUSUM‐type testing procedure (of Berkes I., Horváth L., Kokoszka P. and Shao Q. 2006. Ann.Statist. 34:1140–1165), requires estimation of the location of a possible change‐point, and then using pre‐ and post‐break subsamples to discriminate between short and long‐range dependence. A simulation study examines the empirical size and power of the Wilcoxon‐type testing procedure in standard cases and with disturbances by outliers. It shows that in standard cases the Wilcoxon‐type testing procedure behaves equally well as the CUSUM‐type testing procedure but outperforms it in presence of outliers. We also apply both testing procedure to hydrologic data.